Quadratic K-theory and Geometric Topology? Bruce Williams U. of Notre Dame [email protected] Introduction Suppose R is a ring with an (anti)-involution : R R, and with choice of central unit such that ¯ = 1: Then one− can! ask for a computa- tion of KQuad(R; ; ); the K-theory of quadratic forms. Let H : KR KQuad(R; ; ) be− the hyperbolic map, and let F : KQuad(R; ; ) K!R be the forget− map. Then the Witt groups − ! H W0(R; ; ) = coker(K0R K0Quad(R; ; )) − −! − F W1(R; ; ) = ker(F : KQuad1(R; ; ) K1R) − − −! have been highly studied. See [6], [29],[32],[42],[46],[68], and [86]{[89]. How- ever, the higher dimensional quadratic K-theory has received considerably less attention, than the higher K-theory of f.g. projective modules. (See however, [39],[35], [34], and [36].) Suppose M is an oriented, closed topological manifold of dimension n: We let G(M) = simplicial monoid of homotopy automorphisms of M; T op(M) = sub-simplicial monoid of self-homeomorphisms of M; and (M) = G(N)=T op(N); S t where we take the disjoint union over homeomorphisms classes of manifolds homotopy equivalent to M: Then (M) is called the moduli space of manifold structures on M: S ? Research partially supported by NSF 2 Bruce Williams In classical surgery theory (see Section 2) certain subquotients of KjQuad(R; ; ) with j = 0; 1; R = Zπ1M; and = 1 are used to com- − ± pute π0 (M): S The main goals of this survey article are as follows: 1. Improve communication between algebraists and topologists concerning quadratic forms 2. Call attention to the central role of periodicity. 3. Call attention to the connections between KQuad(R; ; ) and (M); not − S just π0 (M): 4. StimulateS interest in the higher dimensional quadratic K-groups. The functor which sends a f.g. projective module P to HomR(P; R) induces an involution T on K(R): For any i; j Z and any T -invariant subgroup 2 X X Kj(R); topologists (see Section 4) have defined groups Li (R): The subgroup⊂ X is called the decoration for the L-group. Here are a few properties: 1. Periodicity LX (R) LX (R) i ' i+4 K0 i 2. L (R) W0(R; ; ( 1) ) 2i ' − − K2 i 3. L (R) W1(R; ; ( 1) ) 2i+1 ' − − Kj Oj 1 4. Li (R) Li − (R), where Oj 1 is the trivial subgroup of Kj 1(R) ' − − 5. Rothenberg Sequences If X Y Kj(R); we get an exact sequence ⊂ ⊂ LX (R) LY (R) H^ i(Z=2; Y=X) :::: ···! i ! i ! ! 6. Shaneson Product Formula For all i; j Z; 2 Kj+1 1 Kj+1 Kj L (R[t; t− ]) = L (R) L (R); i+1 ∼ i+1 ⊕ i 1 where we extend the involution on R to the Laurent ring R[t; t− ] by 1 t¯= t− : X Z 1 Notice that Li (R) [ 2 ] is independent of X: Different choices for X are used to study various geometric⊗ questions. The classification of compact topo- logical manifolds uses X K1 (See section 2). The study of open manifolds ⊂ involves X Kj; with j < 1; (See [25],[54],[31],[50],[65] and [71]). The study ⊂ of homeomorphisms of manifolds involves X Kj; with j > 1; (see section 5) . ⊂ Localization Sequences: Suppose S is a multiplicative system in the ring R: Then we get an exact sequence 1 Ki(R; S) Ki(R) Ki(S− R) ; ···! ! ! !··· where Ki(R; S) is the K-theory of the exact category of S-torsion R-modules of homological dimension 1. In the case of L-theory(with appropriate choice Quadratic K-theory and Geometric Topology 3 of decorations) one gets an analogous exact sequence using linking forms on torsion modules (see [61],[64], and [53]). However, what is striking about the L-theory localization is that it is gotten by splicing together two 6-term exact sequences. One of these involves (R; ; +1) quadratic forms and the other involves (R; ; 1) quadratic forms. The− resulting sequence is then 12-fold periodic. In fact− − L-theory satisfies many other such periodic exact sequences (see [64]). < > K0 K 1 Let Li −∞ (R) be the direct limit of Li (R) Li − (R) : Let K(R) be the K-theory spectrum constructed! by Wagoneer!··· [78] where for all i Z;Ki(R) πi(K(R)): Similarly, let KQuad(R; ; ) be the spectrum 2 ' − where for all i Z; KQuadi(R; ; ) πi(KQuad(R; ; )): Similarly, let KHerm(R; ; )2 be the K-theory spectrum− ' for Hermitian− forms (see Sections 1 and 3). There− is a functor Quad(R; ; ) Herm(R; ; ) which induces a homotopy equivalence on K-theory when− 2! is a unit in R:− Given a spectrum K equipped with an action by a finite group G we get the norm homotopy fibration sequence N H (G; K) H∗(G; K) H^ ∗(G; K); ∗ −! ! where H (G; K) is the homotopy orbit spectrum of G acting on K, H∗(G; K) is the homotopy∗ fixed spectrum, and N is the norm map. The key example for us is K = K(R);G = Z=2, and the action is given by the involution T: Theorem 0.1 (Hermitian K-theory Theorem). There exists a homotopy cartesian diagram KHerm(R; ; ) (R; ; ) − −−−−! L − F~? ? ? ? y ^ y H∗(Z=2; K(R)) H∗(Z=2; K(R)) −−−−! with the following properties. 1. If 2 is a unit in R, then for i = 0 or 1; < > πi (R; ; +1) L −∞ (R) L − ' i < > πi (R; ; 1) L −∞ (R) L − − ' i+2 2. Periodicity: If 2 is a unit in R, then Ω2 (R; ; ) (R; ; ): F~ L − 'L − − 3. The composition KHerm(R; ; ) H∗(Z=2; K(R)) K(R) is the for- getful map F . − −! ! 4. The homotopy fiber of KHerm(R; ; ) (R; ; ) is a map H~ : H (Z=2; K(R)) KHerm(R; ; )−such that!L the composition− K(R) ∗ ~ ! − ! H (Z=2; K(R)) H KHerm(R; ; ) is the hyperbolic map. ∗ −! − 4 Bruce Williams We call F~ the enhanced forgetful map, and H~ the enhanced hyperbolic map. Before we state an analogous theorem for (M) we need to introduce some more background. S Let hcob(M) be the simplicial set of h-cobordisms on M and let hcob(M) S(M) be the map which sends an h-cobordism h:(W; @W ) (M I;M @I!) ! × × to h M1 : M1 M 1 where @W = M M1: Let HCOB(M) be the homotopy colimitj of ! × t hcob(M) hcob(M I) hcob(M I2) : ! × ! × ··· Igusa has shown that if M is smoothable, then the map hcob(M) HCOB(M) is at least k+1-connected where n = dimM max(2k+7; 3k+4)!: ≥ Let ΩW H(Zπ1(M)) be the homotopy fiber of the assembly map H∗(M; KZ) KZπ1(M): For n > 4; the s-cobordism theorem yields a bijec- ! tion π0(hcob(M)) π0(ΩW H(Zπ1(M)): Waldhausen [80] and Vogel [76],[77] ! has shown how in the definition of KZπ1(M) we can replace π1(M) with the loop space of M and Z with the sphere spectrum. This yields A(M); the K- theory of the space M: There exists a linearization map A(M) KZπ1(M) which is 2-connected. Furthermore, there exists a homotopy! equivalence HCOB(M) ΩW H(M); where ΩW H(M) is the homotopy fiber of the assembly map!H (M; A( )) A(M): ∗ ∗ ! X X Constructions of Ranicki yield spectrum L (R) such that πi(L (R)) X K0 ' Li (R): Let L be the 1-connected cover of L (Z): Theorem 0.2 (Manifold Structure Theorem). Poincare duality yields an involution on WH(M); and there exists a homotopy commutative diagram (M) −∞(M) S −−−−! S ? ? ? ? y ^ y H∗(Z=2; ΩW H(M)) H(Z=2; ΩW H(M)) −−−−! with the following properties. 1. There exists a homotopy equivalence between −∞(M) and the 1-connected cover of Ωn of the homotopy fiber of theS assembly map − < > H (M; L) L −∞ (Zπ1(M)): ∗ ! (See[47] for background on assembly maps.) 4 < > 2. Periodicity: There exists a map −∞(M) Ω −∞ (M) which in- S ! S duces an isomorphisms on πi for i > 0: 3. The composition π0( (M)) π0(H∗(Z=2; ΩW H(M))) π0(ΩW H(M)) S ! ! sends a homotopy equivalence h1 : M1 M to the Whitehead torsion of ! h1: Quadratic K-theory and Geometric Topology 5 4. The map hcob(M) S(M) factors thru the homotopy fiber of (M) ! S ! −∞(M): 5. IfS M is a smoothable manifold, then th above homotopy commutative diagram is homotopy cartesian thru dimension k + 1 where dimM max(2k + 7; 3k + 4): ≥ Notice the strong analogy between these two theorems. K1(R) In section 1 we use the hyperbolic and forgetful maps to define Li (R) for i Z: The product formula is then used to define LKj (R)(R) for j 0 and 2 K1(R) ≤ (R; ; ) any Hermitian ring. In section 2 we explain how Li (R) is used to classify− manifolds up to homeomorphism. In section 3 we use constructions X of Thomason and Karoubi to define Li (R) for X any Z=2-invariant subgroup of Kj(R) for any i; j Z under the assumption that 2 is a unit in R: This assumption is needed in2 order to be able to use Karoubi Periodicity . In section 3 we also discuss the Hermitian K-theory Theorem. In section 4 we discuss Ranicki's approach to L-theory via structures on chain complexes. This yields a periodicity theorem where we do not have to assume 2 is a unit, and for any X i; j Z and any T -invariant subgroup X Kj(R); we define L (R) where (R; 2; ) is any Hermitian ring.